On the other hand, \(\operatorname{SL}_2(\mathbb {F}_4)\)  does embed into \(\operatorname{PGL}_2(\) , and by a theorem of Tate, all projective Galois representations lift. We can follow the proof given by Serre in [1]} to obtain a lift, unramified away from 653, and with Artin conductor \(653^2\) . The fixed field of the kernel of this representation is a quadratic extension of \(L[x]/(x^4-x^3+82x^2-1102x+13537)\) , which is itself the compositum of \(L\)  and the quartic subfield of the 653rd roots of unity. Locally at 653 it is a faithful representation of \(\operatorname{Gal}(\mathbb {Q}_{653}(\@root 8 \of {653},\sqrt{2})/\mathbb {Q}_{653})\) , a Galois group isomorphic to \(\langle x,y|x^8=y^2=e, yx=x^5y\rangle \) .
